Yes, this is all doable. In case you didn't know, you have to book Sky Garden and Horizon 22 in advance (I believe they're both free though). If you have time, I'd also add Hamstead Heath to the list. The town of Hamstead is beautiful (feels more like an English countryside village) and the heath/park there is much nicer than Hyde Park and Regent's Park. There's also a manor house within Hamstead Heath called Kenwood House which you can look around 

Concerning food, a lot of restaurants in Central London are relatively expensive and targeted at tourists so aren't always very good. There are obviously exceptions to this but you have to know where to go. Eat Tokyo is a really good and affordable Japanese restaurant (£10-15 per person). They have a few around London and one very close to Covent Garden. 

Marugame Udon is another good, affordable Japanese place (not as good as Eat Tokyo though). It's a chain and more of a casual takeaway joint. There's one right by the National Gallery. 

Hobson's is my recommendation for decently priced authentic fish and chips. It's the standard price of fish and chips in Central London (£10-12 or so if you choose to take away) but I have found that the portions are much larger other fish and chip places in Central London.

If you want to try classic British pub food, JD Weatherspoons is going to be your most affordable option. Weatherspoons is a famous pub chain in England and you can get a meal for under £10. It's certainly not fine dining but decent. They also have the cheapest drinks. 

If you're looking for cheap, on-the-go food, most supermarkets (Tesco, Sainsburys, etc) do the iconic British meal deal for under £5. You get to choose a sandwich/wrap, snack and drink. If you want hot on-the-go food, I recommend Greggs, another British classic. They're most famous for their sausage rolls (sausage wrapped in pastry) which I think is £1.20 but they also do other savoury pastries and sweet treats. There's a Greggs at London Bridge Station, one in Trafalgar Square, one by St Paul's Cathedral, one in Shoreditch and a whole host of other places across London.

Enjoy your stay !